As we’re at the International Blues Challenge this week, it follows that this week’s Science Wednesday is about music!
Pitch is based on frequency relationships among sounds. The basic pitch structure is an interval, or a pair of frequencies presented melodically (successively) or harmonically (simultaneously). Pythagoras, the mathematician, recognized that some intervals are especially useful in music and pairs of frequencies can be reduced to small, whole-number fractions. For example:
2:1 — the perfect octave
3:2 — the perfect fifth
4:3 – the perfect fourth
Others include the major and minor thirds, and major (“whole” step) and minor (“half step”) seconds.
A tone with a frequency of 440 Hertz (Hz), for example, is called the note “A.” An octave higher is 880 Hz. So, the perfect octave is 880/440 Hz, or, 2:1. A perfect fifth, or the note called “E” is 660 Hz (440 Hz multiplied by the perfect fifth ratio (3/2) is 660 Hz). The image below shows the frequencies associated with each note.
A chord is a combination of intervals. An analysis of the intervals most frequently used in chords shows that the perfect octave, perfect fifth, perfect fourth (in some contexts), major third, and minor third are favored. With each of these intervals, the simultaneous sounding of frequencies is regarded by many listeners as pleasant. These intervals are identified as consonant intervals, and they combine well to form chords (aka, harmonies).
The second (major or minor) is an interval that many listeners do not regard as pleasant when the pair of frequencies in the interval are presented simultaneously. Major and minor seconds are among those identified as dissonant intervals. However, an analysis of the intervals used most frequently in melodies shows that seconds (aka, “steps”) predominate. The dissonance is mitigated by the fact that the intervals are not presented simultaneously (“harmonically”) but successively (“melodically”) instead.
From a scientific standpoint, by far the most important characteristic of these simple consonant intervals is that they are defined by ratios of small integers. Imagine – Pythagoras had invented integers and arithmetic. But it was just abstract thought. He had no way of knowing whether mathematics would be useful for anything. It then turns out that integers and ratios could describe consonant musical intervals. This highlights a deep connection between the human experience and abstract mathematics. What appears to be a subjective judgment – that this interval sounds good (consonant), that interval sounds bad (dissonant) – can be predicted using abstract mathematics. Apparently, our emotional response to the world sometimes even follows mathematical laws!
photo: by Ulyana Peña